48-175 descriptive geometry
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48-175 Descriptive Geometry
Spatial Relations on Lines
1
Lines parallel to a plane
A line is parallel to a plane if it has no common point with the plane.
To test whether a given line and plane are parallel:
simply, construct an edge view of the plane and project the line into the same view; if the line appears in point view or parallel to the edge view, then it cannot meet the plane in a point, and is therefore parallel to the plane
This fact can be used to construct a plane parallel to a given line or a line parallel to a given plane.
2
1
M
B
C
A
O
O
C
A
B
1 3
Line through O parallel to theedge view of plane ABC
2
1
M
O
B
A
C
B
C
A
O
O
C
A
B
1 3
Two possible lines parallel toplane ABC through point O
Line through O parallel to theedge view of plane ABC
2
1
M
O
B
A
C
B
C
A
O
O
C
A
B
How do we locatepoints M and N?
1 3
Two possible lines parallel toplane ABC through point O
Line through O parallel to theedge view of plane ABC
2
1
M
N
M
O
B
A
C
B
C
A
O
O
C
A
B
M
N
How do we locatepoints M and N?
1 3
Two possible lines parallel toplane ABC through point O
Line through O parallel to theedge view of plane ABC
2
1
M
N
N
M
O
B
A
C
B
C
A
O
O
C
A
B
M
N
line parallel through a point parallel to a plane
parallel planes
How do we determine if two planes are parallel ?
2
1
D
E
E
F
F
B
C
A
D
C
A
B
by constructing an auxiliary view that shows
one plane in edge view; if the other plane is
also seen in edge view then the two planes
are parallel
parallel planes
Constructing an auxiliary view that shows one plane in edge view; if the other plane is also seen in edge view then the two planes are parallel
1 3
2
1
D
E
E
F
F
C
B
B
C
A
D
C
A
B
E
parallel edge viewsindicate parallel planes
1 3
2
1
D
D
E
E
F
F
F
C
A
B
B
C
A
D
C
A
B
E
line perpendicular to plane (normal)
A line is perpendicular to a plane if every line in the plane that passes through the point of intersection of the given line and the plane makes a right angle with the given line Line AB is perpendicular to
the plane
Lines LM, NO, PQ alllie in the plane
B
A
P
Q
L M
N
O
perpendicular line to plane (normal)
p
90º90º
32
2
1
p is a planeLine AB is a normal to itLines LM, NO and PQ lie in the plane
A
BM,NL,O
B M,QL,P
P QA,B
L
MN
O
A
direction of the normal to a plane
normal is perpendicularto the edge view of plane
2
1
direction of thenormal in view #2
TL
normal is perpendicularto the edge view of plane
1 3
2
1
direction of thenormal in view #1
90º
direction of thenormal in view #2
TL
normal is perpendicularto the edge view of plane
1 3
2
1
direction of thenormal in view #1
90º
direction of thenormal in view #2
direction of the normal to a plane
direction of thenormal in view #2
direction of thenormal in view #1
2
1
TL
direction of thenormal in view #2
90º
direction of thenormal in view #1
1
2
TL
TL
2
1
direction of thenormal in view #2
direction of thenormal in view #1
90º
90º
Two-view method to find direction (bearing)
quiz: perpendicular to the plane at point P
1
2
B
A
C
P
A
C
B
1
2
P
B
A
C
P
A
C
B
1
2
P
B
A
C
P
A
C
B
1
2
P
B
A
C
P
A
C
B
shortest distance from a point and a plane
Observer's line of sight – plane ABC is seen as an edgeand true length of OP appears
Shortest line (OP) frompoint O to plane ABC
Lines AD and EFlie in the plane ABC
P
A
C
B
O
D
E
F
2
1
C
A
A
C
Edge view of plane ABC
True length of the shortestline from M to the plane
2
1
13
P
M
C
B
MC
A
B
M
A
C
Bh
h
Edge view of plane ABC
True length of the shortestline from M to the plane
P is located by using thetransfer distance from view #3or by tracing a line on theplane through P
2
1
13
P lies on the perpendicular from Mto the true length line in view #1
P
P
P
M
C
A
B
MC
A
B
M
A
C
Bh
h
Edge view of plane ABC
True length of the shortestline from M to the plane
P is located by using thetransfer distance from view #3or by tracing a line on theplane through P
2
1
13
P lies on the perpendicular from Mto the true length line in view #1
P
P
P
M
C
A
B
MC
A
B
M
A
C
B
perpendicular planes
this requires finding edge views of the plane and seeing if they are perpendicular to each other – which we will consider it later when we
consider lines of intersection
how do we determine if a plane is perpendicular to a given plane ?
revisiting an old problem – shortest distance to a line
constructing shortest distance to a line (line method)
As line AB is in true length, the constructedperpendicular from X to AB produces point Y
True length of theshortest distance
1
2
A
X
A
BX
TL
As line AB is in true length, the constructedperpendicular from X to AB produces point Y
True length of theshortest distance
31
1
2
X
A
B
A
X
A
BX
TL
4 3
As line AB is in true length, the constructedperpendicular from X to AB produces point Y
Point view of line AB
True length of theshortest distance
31
1
2
X
AB,Y
X
A
B
A
X
A
BX
TL
Project back from view #1 to get Y
Project back from view #3 to get Y
4 3
As line AB is in true length, the constructedperpendicular from X to AB produces point Y
Point view of line AB
True length of theshortest distance
31
1
2
Y
Y
Y
X
AB,Y
X
A
B
A
X
A
BX
constructing shortest distance to a line (plane method)
ABX defines a plane
X Y is the shortestdistance from X to AB
1
2
A
B
X
A
BX
ABX defines a plane
X Y is the shortestdistance from X to AB
Edge view of ABX
31
1
2
A
X
B
A
B
X
A
BX
ABX defines a plane
X Y is the shortestdistance from X to AB
43
Edge view of ABX True shape of ABX3
1
1
2
A
XX
B
A
B
X
A
BX
ABX defines a plane
X Y is the shortestdistance from X to AB
Project back from view #4 to get Y
43
Project back from view #1 to get Y
Edge view of ABX True shape of ABX3
1
1
2
A
XX
B
A
B
X
A
BX
shortest distance between skew lines
A
B
D
C
Line XY is the shortest distancebetween skew lines AB and CDas it is perpendicular to both lines
90°
90°X
Y
shortest distance between skew lines
2
1
A
B
C
D
B
A
D
C
shortest distance between skew lines (line method)
90°
90°
Common perpendicular XYbetween skew lines AB andCD in view #1
AB is in true length in view #3
Common perpendicular XY betweenskew lines AB and CD in view #2
True length of shortest lineXY is seen in view #4
4 3
3
1
2
1
X
Y
X
X
Y
Y
YD
C
AB, X
CD
AB
A
B
C
D
B
A
D
C
shortest distance between skew lines (plane method)
2
1
A
B
C
D
B
A
D
C
HL2
1
Y
X
W
WX
A
B
C
B
A
D
C
shortest distance between skew lines (plane method)
HL
43
Shortest distance RS between skew lines AB and CD
Plane CXYD seen inedge view in view #3
13
CXYD is a plane
XY is parallel to AB in view #1and passes through W
XY is parallel to AB in view #2and meets CD at WDY is parallel to folding line 1|2
2
1
S
R
S
RS
R
R,S
C
D
B
A D, Y
C,X
A
B
Y
X
W
WX
Y
A
B
C
D
B
A
D
C
shortest horizontal distance between skew lines
parallel
Shortest horizontal distancebetween the two skew lines
Horizontal projection plane
XY
Y
X
shortest horizontal distance between skew lines
1
2
B
A
D
C
A
B
D
C
shortest horizontal distance between skew lines
HL
TL
View #3 is an elevation
XY is parallel to the edge view ofthe horizontal planeXY is also the true length of theshortest horizontal line
CD is parallel to the edgeview of plane ABLM inview #3
BL is intrue length
LM is parallel to CDin view #1
LM is parallel toCD in view #2
2
1
M
A
D
CB,L
L
M
L
MC
A
B
A
B
D
C
shortest horizontal distance between skew lines
HL
TL
View #3 is an elevation
XY is parallel to the edge view ofthe horizontal planeXY is also the true length of theshortest horizontal line
3
CD is parallel to the edgeview of plane ABLM inview #3
4
1
3
BL is intrue length
LM is parallel to CDin view #1
LM is parallel toCD in view #2
2
1
M
X Y
Y
X
Y
X
X,Y
B
C
A
D
A
D
CB,L
L
M
L
MC
D
A
B
A
B
D
C
shortest grade distance between skew lines
HL
TL
15°
View #3 is an elevation
XY is also the true length of the shortest upward 15° grade line
3
4
1
3
BL is intrue length
LM is parallel to CDin view #1
LM is parallel toCD in view #2
2
1
Y
X
Y
X
Y
X
X,Y
B
C
D
A
O4
M
A
D
CB,L
L
M
L
MC
D
A
B
A
B
D
C
N4
which grade distance?
HL
TL
90°
20% grade
View #3 is an elevation
XY is also the true length of theshortest downward 20%grade line
3
4
1
3
BL is intrue length
LM is parallel to CDin view #1
LM is parallel toCD in view #2
2
1
XY
Y
X
Y
X
C
DB
A
M
A
D
CB,L
L
M
L
MC
D
A
B
A
B
D
C
visibility
Observers line of sight in whichline AB is above line CD
D
CB
A
quiz: find a point on a line equidistant to two points
l
l
f
t
A
B
B
A
l
l
l
f
t
B
A
A
B
B
A
l
l
TL
l
Project back from top view to get X
Project back from view #1 to get X
1t
f
t
X
X
X
midpoint
B
A
A
B
B
A
l
l
TL
l
1t
f
t
X
X
midpoint
B
A
A
B
B
A
quiz: locating a line between two skew lines through a point
quiz: construct a line at a certain grade
Lines AB and CD specify centerlines of two existing sewers as shown in the figure. The sewer pipes are to be connected by a
branch pipe having a downward grade of
2:7 from the higher to the lower pipe. Given that point C is 20' North of point A, the problem is to determine the true length and bearing of the branch pipe and show this pipe in all views. Line AC (in plan) measures 20'.
20'
f
t
Y
X
Y
C
D
B
D
C
A
A
B
20'
1t
XD
C
B,Y
A,X
Y
C
D
BA
TL = 30'-8"
bearing = S 51.5° E
21
2:7 grade
D
C
B
AD
C
B,Y
A,X
20'
bearing = S 51.5° E
f
t
X
Y
Y
C
D
B
D
C
A
A
B
quiz: construct a line at a certain grade
quiz: shortest distance from X to nearest face
5'-0"4'-0"
3'-6"
1'-6"
1'-0"
45°
X
X
a
f
t
X
X
a
edge view of face a
t1
f
t
X1
X
X
Y is not on face p∴ need to find point on face anearest Y in the same plane
Y is the foot of the perpendicularfrom X to the plane of face p
p
edge view of face p
t1
f
t
Y
Y
X
X
X
p
Z is projected back from view #1
Z is projected back from view #2
true length of ABZ is perpendicular from XY to AB in view #2
2
1
t
1
f
t
Z
Z
Z
X,YA
Y
Y
B
A
X
X
X
AB
= 2'-1"
true shortest distance betweenX and Z is given in view #3
3t
p
2
1
t1
f
t
Z
X
Z
Z
Z
X,YA
Y
Y
B
A
X
X
X
AB
Problem
Three equal legs of a surveyor’s tripod are located in their relationship to the plumb line.Leg A bears N30°W and has a slope of 30°Leg B is 3’-3” due east of the plumb line and at the same elevation as the plumb lineLeg C bears S45°W and has a slope of 45°The plumb bob touches the bench mark at a vertical distance of 4’ below the top of the line
Determine TL of legs A, B and C ?
What is the angle B makes with plumb line?
Show legs in front and top views.
Step 1
Step 2
Problem
Two sewer lines AB and CB converge at manhole BA is 35’ north 10’ east of B and 30’ above BC is 20’ north 60’ west of B and 15’ above BA new line DB is located in the plane of ABC at a point 30’ due west of A
Using only two views locate D
What is the TL of each sewer line?
What is the angle of the plane ABC
Step 1
Step 2
Step 3
Problem AB is parallel to plane. Complete the view.
Step
Problem
Planes ABC and DEF
are parallel.
Complete the view
Step
What is the shortest length between non-
intersecting diagonals of adjacent faces of a cube?
Problem
Given two partial pipelines –
construct the shortest level pipeline to connect them.
These pipelines may be extended
Step 1
Step 2
Step 3
Problem involving perpendicular lines
Construct the pyramid with
base A(2, 2, 5), B(3.5, 1.5, 6), C(4.75, 2, 6) D(3.25, 2.5, 5)height 2.5
All measurements in inchesComplete top and front views with proper visibility ie., visible lines are
solid and not visible lines shown dashed.
What is the clearance between a pipe and the ball?
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